Compatibility of the $L^2$-pure Hodge structure with Saito's pure Hodge structure on intersection cohomology (Shentu–Zhao, Conjecture 8.5)

For a compact Kähler space $X$ carrying a polarizable variation of Hodge structures $\mathbb{V}$ with quasi-unipotent local monodromies on a suitable open part $X^o$, Shentu and Zhao (arXiv:2103.04030) constructed complete Kähler metrics of controlled asymptotics near the singular strata ('Kähler distinguished metrics') and proved that the $L^2$-cohomology of square-integrable forms with coefficients in $\mathbb{V}$ is canonically isomorphic to the intersection cohomology $IH^*(X,\mathbb{V})$ (their Theorem 1.4). Because the metric is Kähler, the $(p,q)$-decomposition of $L^2$-forms endows this $L^2$-cohomology with a pure Hodge structure, while Saito's theory of Hodge modules endows $IH^*(X,\mathbb{V})$ with a purely algebraic pure Hodge structure. Their Conjecture 8.5 asserts that these two pure Hodge structures coincide under the canonical isomorphism. Only two partial cases are known: the case of zero-dimensional $X\setminus X^o$ with trivial $\mathbb{V}$ (by an argument going back to Zucker) and the single extreme Hodge piece (their Theorem 8.6, via the companion S-sheaf paper). Shentu–Zhao's Conjecture 8.5 remains open.

Background

The interaction between the topology of singular complex varieties and analysis on their smooth loci began with Cheeger, Goresky and MacPherson (1982), who conjectured that the $L^2$-cohomology of the regular locus of a projective variety, equipped with the restriction of the Fubini–Study metric (the standard Kähler metric on complex projective space), recovers the intersection cohomology of the variety. Intersection cohomology, invented by Goresky and MacPherson, is a homology theory for singular spaces that restores Poincaré duality by restricting which chains count - the middle-perversity condition bounds how allowable chains may meet the singular strata; $L^2$-cohomology is the cohomology of the complex of differential forms that are square-integrable together with their differentials with respect to a chosen complete Riemannian metric. This original 'standard metric' conjecture is confirmed only in special cases - normal surfaces (Hsiang–Pati 1985) and isolated singularities (Ohsawa 1991) - and remains open in general. Independently, Saito's theory of Hodge modules (1988) - a vast generalization of variations of Hodge structure to singular spaces, built from filtered $\mathcal{D}$-modules (modules over sheaves of differential operators) carrying a rational structure - endows the intersection cohomology $IH^*(X,\mathbb{V})$ with a pure Hodge structure for any polarizable variation of Hodge structures $\mathbb{V}$. Here a variation of Hodge structures is a locally constant sheaf of complex vector spaces on the smooth locus whose fibers carry Hodge decompositions varying holomorphically (for example the local systems underlying a smooth family of projective varieties), polarizable means equipped with a positive definite Hodge metric $h$, and a pure Hodge structure of weight $k$ on a rational vector space is a decomposition of its complexification into pieces of type $(p,q)$ with $p+q=k$. Shentu and Zhao (2021) developed an $L^2$-realization of this algebraic theory: for a compact Kähler space $X$ and a dense Zariski open subset $X^o$ of the regular locus $X_{\mathrm{reg}}$, they construct a complete Kähler metric $ds^2$ on $X^o$ with prescribed asymptotics along the singular strata - a Kähler distinguished metric - and prove (Theorem 1.4) that the $L^2$-cohomology $H^*_{(2)}(X^o,\mathbb{V};ds^2,h)$, defined with respect to $ds^2$ and the Hodge metric $h$, is canonically isomorphic to $IH^*(X,\mathbb{V})$ as a graded vector space. Since $ds^2$ is Kähler, the $L^2$-forms admit a refinement of the Hodge decomposition into $(p,q)$-types, giving the left-hand side a pure Hodge structure (the paper's title, '$L^2$-representation of Hodge modules', refers to this differential-geometric realization of Hodge-module cohomology). The authors verify compatibility with Saito's Hodge structure in two situations: when $X\setminus X^o$ consists of isolated points and $\mathbb{V}$ is trivial (via Zucker's 1987 study of isolated singularities), and for the single extreme Hodge piece (their Theorem 8.6, whose proof relies on their companion paper resolving the lowest Hodge piece through an 'S-sheaf' Dolbeault resolution, published in Advances in Mathematics in 2024). The general compatibility statement is left as their Conjecture 8.5, the target of this problem.

Problem Statement

Let $X$ be a compact Kähler space (a compact complex space whose regular locus carries a Kähler metric), let $X^o\subset X_{\mathrm{reg}}$ be a dense Zariski open subset of the regular locus, and let $\mathbb{V}$ be a polarizable variation of Hodge structures on $X^o$ with quasi-unipotent local monodromies and Hodge metric $h$ (the source states its theorem for the strictly more general class of 0-tame harmonic bundles, which by their Remark 4.22 contains the underlying hermitian local system of an $\mathbb{R}$-polarized variation of Hodge structures with quasi-unipotent local monodromies). Let $ds^2$ be a Kähler distinguished metric on $X^o$: a complete Kähler metric associated to the pair $(X,X^o)$ in the sense of Definition 4.12 of Shentu–Zhao, arXiv:2103.04030, complete with locally finite volume and with the prescribed asymptotic behavior along $X\setminus X^o$ used in their Theorem 1.4. By that theorem there is a canonical isomorphism of graded vector spaces

$$H^*_{(2)}(X^o,\mathbb{V};ds^2,h)\ \cong\ IH^*(X,\mathbb{V}),$$

where $H^*_{(2)}(X^o,\mathbb{V};ds^2,h)$ denotes the $L^2$-cohomology of square-integrable forms on $X^o$ with coefficients in $\mathbb{V}$, and $IH^*(X,\mathbb{V})$ denotes the intersection cohomology of $X$ with coefficients in $\mathbb{V}$. Because $ds^2$ is Kähler, the $(p,q)$-decomposition of $L^2$-forms endows the left-hand side with a pure Hodge structure, while Saito's theory of polarizable Hodge modules endows the right-hand side with a pure Hodge structure. Shentu–Zhao's Conjecture 8.5 asserts that these two pure Hodge structures coincide under the canonical isomorphism above. Resolve this conjecture: prove the compatibility for every compact Kähler space, dense Zariski open subset, polarizable variation of Hodge structures with quasi-unipotent local monodromies, and Kähler distinguished metric as above, or disprove it by an explicit counterexample.

References

1. Junchao Shentu, Chen Zhao, '$L^2$-representation of Hodge Modules', preprint, 2021, arXiv:2103.04030, https://arxiv.org/abs/2103.04030
2. Junchao Shentu, Chen Zhao, 'L2-Dolbeault resolution of the lowest Hodge piece of a Hodge module', Advances in Mathematics 442, 2024, article 109547, DOI 10.1016/j.aim.2024.109547, https://doi.org/10.1016/j.aim.2024.109547
3. Junchao Shentu, Chen Zhao, '$L^2$-extension of adjoint bundles and Kollár's conjecture', Mathematische Annalen 386, 2023, pp. 1429–1462, DOI 10.1007/s00208-022-02432-6 (preprint arXiv:2106.13407), https://doi.org/10.1007/s00208-022-02432-6
4. Mads Bach Villadsen, 'Hodge modules and Kähler morphisms', preprint, 2024, arXiv:2401.09544, https://arxiv.org/abs/2401.09544
5. Chen Zhao, '$L^2$ representation of Simpson-Mochizuki's prolongation of Higgs bundles and the Kawamata-Viehweg vanishing theorem for semistable parabolic Higgs bundles', preprint, 2022, arXiv:2210.04130, https://arxiv.org/abs/2210.04130
6. Chen Zhao, 'Minimal extension property of direct images', preprint, 2024, arXiv:2409.04754, https://arxiv.org/abs/2409.04754
7. Jeff Cheeger, Mark Goresky, Robert MacPherson, '$L^2$-cohomology and intersection homology of singular algebraic varieties', in Seminar on Differential Geometry (Annals of Mathematics Studies 102), Princeton University Press, 1982, pp. 303–340, DOI 10.1515/9781400881918-018, https://doi.org/10.1515/9781400881918-018
8. W. C. Hsiang, V. Pati, 'L2-cohomology of normal algebraic surfaces. I', Inventiones Mathematicae 81, 1985, 395–412, DOI 10.1007/BF01388578, https://doi.org/10.1007/BF01388578
9. Steven Zucker, 'The Hodge structures on the intersection homology of varieties with isolated singularities', Duke Mathematical Journal 55, 1987, 603–616, DOI 10.1215/s0012-7094-87-05530-x, https://doi.org/10.1215/s0012-7094-87-05530-x
10. Takeo Ohsawa, 'Cheeger-Goreski-MacPherson's conjecture for the varieties with isolated singularities', Mathematische Zeitschrift 206, 1991, 219–224, DOI 10.1007/BF02571337, https://doi.org/10.1007/BF02571337
11. Michael Harris, Steven Zucker, 'Boundary cohomology of Shimura varieties. III: Coherent cohomology on higher-rank boundary strata and applications to Hodge theory', Mémoires de la Société mathématique de France, Nouvelle Série 85, 2001, DOI 10.24033/msmf.398, https://doi.org/10.24033/msmf.398
12. Morihiko Saito, 'Modules de Hodge polarisables', Publications of the Research Institute for Mathematical Sciences 24, 1988, pp. 849–995, DOI 10.2977/prims/1195173930, https://doi.org/10.2977/prims/1195173930
13. Jiaming Luo, Shirong Li, 'Stratified Interpretation for De Rham Cohomology and Non-Witt Spaces', preprint, 2025, arXiv:2505.00320, https://arxiv.org/abs/2505.00320
14. Jiaming Luo, Shirong Li, 'Derived Stratifications and Arithmetic Intersection Theory for Varieties with Isolated Singularities', preprint, 2025, arXiv:2508.01679, https://arxiv.org/abs/2508.01679
15. Jiaming Luo, 'Derived Stratified-Microlocal Framework and Moduli Space Resolution for the Cheeger-Goresky-Macpherson Conjecture', preprint, 2025, arXiv:2508.17833, https://arxiv.org/abs/2508.17833